Prior to this module the following topics have been investigated: 1) functions, 2) function classes, 3) average rate of change, 4) limits, 5) difference equations and 6) how different function classes are characterized by average rate of change, percent change, limiting behavior and difference equation models.
This module begins approximately three weeks into the course and takes approximately 4-5 weeks to complete. In the activities in this module:
Students develop and apply a definition of instantaneous rate of change and the derivative by moving from average rate of change estimates to an exact rate of change at an instant.
Students investigate typical application problems (optimization, related rates etc) as soon as the first derivative is discovered. This encourages students to both deepen their understanding of the meaning of the derivative and develop their own independent approaches to these types of problems.
Students conduct mathematical investigations to conjecture rules of differentiation.
Students begin developing differential equation models for the first time (building from the understanding of difference equations developed in Module I).
Module II: The Derivative